Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 342 / 3 / c

Table of contents
- Solution c

Solution

Let

a

denote tube cross-sectional area in this question, not a filament or sphere radius. The bacterial density is uniform across that section in this one-dimensional Keller--Segel model, so the total bacterial number is

N = a \int_{- \infty}^{\infty} B (z) d z .

(1)

Integrating the nutrient balance

v C^{'} = k B

across the complete band gives

v [C (+ \infty) - C (- \infty)] = k \int_{- \infty}^{\infty} B (z) d z = \frac{k N}{a} .

(2)

Hence the speed of a nutrient-consuming chemotactic band is

v = \frac{N k}{a C _{\infty}} .

(3)

This is a nutrient budget: the whole population consumes at rate

N k

, while advancement of the band at speed

v

brings fresh nutrient at rate

a v C_{\infty}

. The conservation of mass equation for bacteria ensures

N

stays constant when there is no flux at infinity and no cell birth or death.

At fixed total bacterial number, the speed grows linearly with

N

and the per-cell consumption rate

k

, and decreases with tube area and nutrient concentration ahead. More nutrient takes longer to deplete, giving the inverse dependence on

C_{\infty}

. Although

D

and

α

determine the width, skewness, peak density and admissibility of the smooth band, they do not appear in this speed when

N, k, a, C_{\infty}

are fixed. This independence relies on neglecting nutrient diffusion and using the specified concentration-independent consumption law; it is not a general prediction for all chemotaxis models. Dimensionally